A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor $D$, while $D$ paired-dominating a~dominating and the subgraph induced by contains perfect matching. graph $D\!P\!D\!P$-graph it pair $(D,P)$ disjoint sets vertices such that $P$ $G$. The study $D\!P\!D\!P$-graphs was initiated Southey Henning (Cent. Eur. J. Math. 8 (2010) 459--467; Comb. Optim. 22 (2...

OF THE DISSERTATION Applications and Variations of Domination in Graphs by Paul Andrew Dreyer, Jr. Dissertation Director: Fred S. Roberts In a graph G = (V, E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications...

We show in this paper that the upper minus domination number −(G) of a claw-free cubic graph G is at most 1 2 |V (G)|. © 2006 Published by Elsevier B.V.

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...

We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n + O (n/g) < 3n/10 + O (n/g).

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that...

We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic graphs. The APX-...

We prove that connected cubic graphs of order n and girth g have domination number at most 0.32127n+O ( n g ) .